On the Absolutely Continuous Subspace for Non - Selfadjoint Operators

Abstract

The paper is devoted to the problem of absolutely continuous and singular subspaces of non – selfadjoint operators defined as pseudo–Friedrichs extensions of operators of the form $\tilde L = A_0 + V_0 + {i \over 2} \alpha \kappa \alpha$ with the natural domain dom $(\tilde L) = {\rm dom} (A_0) \cap {\rm dom} (V_0) \cap {\rm dom} (\alpha \kappa \alpha)$, where $A_0$, $V_0$, $\alpha$, $\alpha \geq 0$, and $\kappa$, $\parallel \kappa \parallel \leq 1$, are self–adjoint operators such that $\sqrt {\mid V_0 \mid}$ and $\alpha$ are relatively bounded with respect ro $\sqrt {\mid A_0 \mid}$. Both subspaces are characterized in the so – called “weak” sense. Model theory for dissipative and non – selfadjoint operator is avoided. Technical tools, which are used, are dilation theory for maximal dissipative operators, Lax – Phillips scattering theory and stationary scattering theory for self – adjoint and dissipative operators. In particular, it is shown that the absolutely continuous subspace of the pseudo – Friedrichs extension is non – trivial if and only if this is true for the self – adjoint pseudo – Friedrichs extension of $A_0 + V_0$. The results are applied to Schrödinger operators with complex potentials.